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A091236 Nonprimes of form 4k+3.

%I A091236 #28 Jul 20 2025 11:50:01
%S A091236 15,27,35,39,51,55,63,75,87,91,95,99,111,115,119,123,135,143,147,155,
%T A091236 159,171,175,183,187,195,203,207,215,219,231,235,243,247,255,259,267,
%U A091236 275,279,287,291,295,299,303,315,319,323,327,335,339,343,351,355,363
%N A091236 Nonprimes of form 4k+3.
%C A091236 If we define f(n) to be the number of primes (counted with multiplicity) of the form 4k + 3 that divide n, then with this sequence f(a(n)) is always odd. For example, 95 is divisible by 17 and 99 is divisible by 3 (twice) and 11. - _Alonso del Arte_, Jan 13 2016
%C A091236 Complement of A002145 with respect to A004767. - _Michel Marcus_, Jan 17 2016
%C A091236 With the Jan 05 2004 Jovovic comment on A078703: The number of 1 and -1 (mod 4) divisors of a(n) are identical. Proof: each number 3 (mod 4) is trivially not a sum of two squares. The number of solutions of n as a sum of two squares is r_2(n) = 4*(d_1(n) - d_3(n)), where d_k(n) is the number of k (mod 4) divisors of n. See e.g., Grosswald, pp. 15-16 for the proof of Jacobi. - _Wolfdieter Lang_, Jul 29 2016
%D A091236 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.
%H A091236 Vincenzo Librandi, <a href="/A091236/b091236.txt">Table of n, a(n) for n = 1..1000</a>
%e A091236 27 = 4 * 6 + 3 = 3^3.
%e A091236 35 = 4 * 8 + 3 = 5 * 7.
%e A091236 a(8) = 75 with 2*A078703(19) = 6 divisors [1, 3, 5, 15, 25, 75], which are 1, -1, 1, -1, 1, -1 (mod 4). - _Wolfdieter Lang_, Jul 29 2016
%p A091236 A091236 := proc(n)
%p A091236     option remember  ;
%p A091236     local a;
%p A091236     if n = 1 then
%p A091236         15 ;
%p A091236     else
%p A091236         for a from procname(n-1)+1 do
%p A091236             if not isprime(a) and modp(a,4) = 3 then
%p A091236                 return a;
%p A091236             end if;
%p A091236         end do;
%p A091236     end if;
%p A091236 end proc:
%p A091236 seq(A091236(n),n=1..20) ; # _R. J. Mathar_, Jul 20 2025
%t A091236 Select[Range[1000], !PrimeQ[#] && IntegerQ[(# - 3)/4] &] (* _Harvey P. Dale_, Aug 16 2013 *)
%t A091236 Select[4Range[100] - 1, Not[PrimeQ[#]] &] (* _Alonso del Arte_, Jan 13 2016 *)
%o A091236 (PARI) lista(nn) = for(n=1, nn, if(!isprime(k=4*n+3), print1(k, ", "))); \\ _Altug Alkan_, Jan 17 2016
%Y A091236 Cf. A002145, A004767, A078703.
%Y A091236 Cf. A091113-A092256.
%K A091236 easy,nonn
%O A091236 1,1
%A A091236 _Labos Elemer_, Feb 24 2004